Codes over Trees Lev Yohananov and Eitan Yaakobi Technion Israel - - PowerPoint PPT Presentation

Codes over Trees

Lev Yohananov and Eitan Yaakobi Technion – Israel Institute of Technology 2020 IEEE International Symposium on Information Theory

Motivation

• Trees and their properties are very beneficial in numerous applications.
• In biology.
• In chemistry.
• In programming languages.
• In cyber applications.
• etc.
• In coding theory, a novel family of codes is presented.

1

Trees

• A finite undirected tree over 𝑜 nodes is a connected undirected graph

with 𝑜 − 1 edges.

• 𝑈 𝑜 : the set of all trees over 𝑜 nodes.
• By Cayley’s formula1 it holds that |𝑈 𝑜 | = 𝑜𝑜−2.

1 2 3 4 5 6 7 8 9

1 M. Aigner and G. M. Ziegler, Proofs from THE BOOK, pp. 141–146,Springer-Verlag, New York, 1998.

2

Codes (over Trees)

• 𝐷𝑈: a code (over trees) denoted by 𝑜, 𝑁 𝑈, such that
• 𝑜: the number of nodes in a tree.
• 𝑁: the size of 𝐷𝑈.
• Example 𝑜 = 5, 𝑁 = 8

2 1 3 1 2 3 2 1 3 3 1 2 4 4 4 4 1 2 3 1 2 3 2 1 3 3 1 2 4 4 4 4

3

Edge Erasure

• An edge erasure is a removal of an edge from a tree.

4

Tree Distance

• Given two trees 𝑈

1 = (𝑊 𝑜, 𝐹1) and 𝑈2 = 𝑊 𝑜, 𝐹2 .

• 𝑒𝑈 𝑈

1, 𝑈2 : the tree distance (or distance) between 𝑈 1 and 𝑈2 is

𝑒𝑈 𝑈

1, 𝑈2 = 𝑜 − 1 − |𝐹1 ∩ 𝐹2|.

• This distance is a metric.

𝑒𝑈 𝑈

1, 𝑈2 = 8 − 7 = 1.

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 5

Codes with Minimal Distance 𝑒

• 𝑜, 𝑁, 𝑒 𝑈: a code over trees of minimal distance 𝒆.
• 𝑠: the redundancy of the code 𝐷𝑈.
• 𝑠 =

𝑜 − 2 log 𝑜 − log(𝑁).

• 𝐵 𝑜, 𝑒 : the largest size of a code of distance 𝑒.
• 𝑠 𝑜, 𝑒 : the minimal redundancy of a code of distance 𝑒.
• Theorem: A 𝑜, 𝑁 𝑈 code over trees 𝐷𝑈 is of tree distance 𝒆 if and
• nly if it can correct any 𝒆 − 𝟐 edges.

6

Forests

• An undirected graph that consists of only disjoint union of trees is

called a forest.

• ℱ 𝑜, 𝑢 : the set of all forests over 𝑜 nodes with exactly 𝒖 trees.
• Note that ℱ(𝑜, 1) = 𝑈(𝑜).

1 2 3 4 5 6 7 8 9 7

Number of Forest with Exactly 𝑢 Trees

• The value of |ℱ 𝑜, 𝑢 | was shown to be

𝐺 𝑜, 𝑢 = 𝑜 𝑢 𝑜𝑜−𝑢−1 ෍

𝑗=0 𝑢

− 1 2

𝑗 𝑢

𝑗 (𝑢 + 𝑗) 𝑜 − 𝑢 ! 𝑜𝑗 𝑜 − 𝑢 − 𝑗 ! .

• Another representation of it

𝐺 𝑜, 𝑢 = 𝑜𝑜−𝑢 ෍

𝑗=0 𝑢

− 1 2

𝑗 𝑢

𝑗 𝑜 − 1 𝑢 − 1 + 𝑗 𝑢 + 𝑗 ! 𝑜𝑗𝑢! .

• J. Moon2 1970.
• B. Bollobas3 1979.

2 J. W. Moon, Counting labeled trees, 1970. 3 B. Bollobas, Graph Theory: An Introductory Course, Springer-Verlag, New York, 1979.

8

Forest Ball of a Tree

• 𝒬𝑈 𝑜, 𝑢 : the forest ball of a tree 𝑈 of radius 𝑢.
• Note that 𝒬𝑈 𝑜, 𝑢 ⊆ ℱ(𝑜, 𝑢 + 1).

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

𝒬𝑈 𝑜 = 5, 𝑢 = 1 = 𝑈 = Note that |𝒬𝑈 𝑜, 𝑢 | = 𝑜 − 1 𝑢

Regular

9

Sphere Packing Bound

• Theorem: For all 𝑜 ≥ 1 and 1 ≤ 𝑒 ≤ 𝑜, it holds that

𝐵 𝑜, 𝑒 ≤ 𝐺(𝑜, 𝑒)/ 𝑜 − 1 𝑒 − 1 . 𝑈

1

𝑈2

𝐺

𝑒 − 1 10

Corollary

• It was also proved2 that for any fixed 𝑒,

lim

𝑜→∞

𝐺 𝑜, 𝑒 𝑜𝑜−2 = 1 2𝑒−1 𝑒 − 1 ! .

• Thus,

𝐵 𝑜, 𝑒 ≤ 𝐺 𝑜, 𝑒 𝑜 − 1 𝑒 − 1 = 𝑃 𝑜𝑜−1−𝑒 .

2 J. W. Moon, Counting labeled trees, 1970.

11

Results from Sphere Packing Bound

• Correcting 𝒐 − 𝟑 erasures: 𝐵 𝑜, 𝑜 − 1 ≤

𝑜 2 .

• Correcting 𝒐 − 𝟒 erasures: 𝐵 𝑜, 𝑜 − 2 = 𝑃 𝑜2 .
• Correcting 𝒐 − 𝟓 erasures: 𝐵 𝑜, 𝑜 − 3 = 𝑃 𝑜3 .

12

The Results of this Work

• Correcting 𝒐 − 𝟑 erasures: 𝐵 𝑜, 𝑜 − 1 = 𝑜/2 .
• Correcting 𝒐 − 𝟒 erasures: 𝐵 𝑜, 𝑜 − 2 = 𝑜.
• Correcting 𝒐 − 𝟓 erasures: 𝐵 𝑜, 𝑜 − 3 = 𝑃 𝑜2 .
• For fixed 𝑒 and 𝑜 ≥ 2𝑒,

Ω 𝑜𝑜−2𝑒 ≤ 𝐵 𝑜, 𝑒 ≤ 𝑃 𝑜𝑜−1−𝑒 . 𝐵 𝑜, 𝑜 − 1 ≤ 𝑜 2 . 𝐵 𝑜, 𝑜 − 2 = 𝑃 𝑜2 . 𝐵 𝑜, 𝑜 − 3 = 𝑃 𝑜3 .

13

Lower Bound on 𝐵(𝑜, 𝑜 − 1)

• A line tree 𝑈:
• Our code will be constructed from 𝑜/2 line trees as follows:
• Thus, this code is a set of 𝑜/2 disjoint Hamiltonian paths, see Lucas4.

1 2 3 4 1 7 2 6 4 3 5 1 2 3 7 5 4 6

𝐵 𝑜, 𝑜 − 1 = 𝑜/2

4 E. Lucas, “Les rondes enfantines,” Recreations mathematiques, vol. 2, Paris, 1894.

14

Lower Bound on 𝐵(𝑜, 𝑜 − 2)

• A star tree 𝑈:
• Our code will be constructed from 𝑜 star trees as follows:
• Every two trees have exactly one edge in common.

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 3 2 4 1 4 2 3

𝐵 𝑜, 𝑜 − 2 = 𝑃 𝑜2 𝐵 𝑜, 𝑜 − 2 ≥ 𝑜

15

Upper Bound on 𝐵(𝑜, 𝑜 − 2)

• Let 𝐻 be a bipartite graph as in example:
• Any code of tree distance 𝑜 − 2 yields a bipartite 𝐻 of girth of at least 6.
• Using Reiman’s inequality5 1958, it is shown that every such bipartite 𝐻

holds 𝑁 ≤ 𝑜. 𝑈

1

𝑈2 ⋮ 𝑈𝑁

𝑓1 𝑓2

⋮

𝑓 𝑜

2

𝐵 𝑜, 𝑜 − 2 = 𝑃 𝑜2 𝐵 𝑜, 𝑜 − 2 ≤ 𝑜

5 I. Reiman, “Uber ein Problem von K. Zarankiewicz,” Acta mathematica hungarica, vol. 9, issue 3–4, pp. 269–273,

Hungary, Budapest, Sep. 1958.

deg egree = 𝒐 − 𝟐

16

Upper Bound on 𝐵(𝑜, 𝑜 − 3)

• Also done using Reiman’s inequality. (short in time)
• In the longest version6: 𝐵 𝑜, 3𝑜/4

= Ω(𝑜2).

• 𝑜 is a prime number.

𝐵 𝑜, 𝑜 − 3 = 𝑃 𝑜3 𝐵 𝑜, 𝑜 − 3 = 𝑃(𝑜2)

6 L. Yohananov and E. Yaakobi, “Codes over trees, ”arXiv:2001.01791,Jan. 2020.

17

Lower Bound on General 𝐵(𝑜, 𝑒)

• Theorem: for fixed 𝑒 and 𝑜 ≥ 2𝑒 it holds

𝐵 𝑜, 𝑒 = 𝛻 𝑜𝑜−2𝑒 .

• Construction: Let (𝑓1, 𝑓2, … , 𝑓 𝑜

2

) be some order of all the edges of the complete graph over 𝒐 nodes.

• Each tree 𝑈 will be represented as a characteristic vector of length

𝑜 2 and weight 𝒐 − 𝟐 as in the example:

1 2 3

1 1 1

{0,1} {0,2} {0,3} {1,2} {1,3} {2,3}

18

The Proof

• A linear binary code of length 𝑂 = 𝑜

2 and Hamming distance 𝐸 = 2𝑒 − 1 can correct at most 𝑒 − 1 substitution.

• Corresponding to 𝑒 − 1 edge erasures.
• Applying BCH codes, we pay the redundancy of

𝑠 = 𝑒 − 1 log 𝑜 2 + 𝑃 1 = 2 𝑒 − 1 log 𝑜 + 𝑃 1 .

• The 2𝑠 cosets of such code are also ( 𝑜

2 , 𝐿, 2𝑒 − 1) codes.

• Thus, by the pigeonhole principle there is a code of cardinality at least

𝑜𝑜−2 22 𝑒−1 log 𝑜 = Ω 𝑜𝑜−2𝑒 .

19

Tree Balls of Trees

20

Tree Ball of Trees

• 𝐶𝑈 𝑜, 𝑢 : tree ball of trees:

|𝒬𝑈 𝑜, 𝑢 | = 𝑜 − 1 𝑢

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

⋯ ⋯ ⋯ ⋯

|𝐶𝑈 𝑜, 𝑢 | =?

Regular Not Regular

𝒬𝑈 𝑜, 𝑢

21

Radius One

22

𝑈 𝑈 𝑈 Θ(𝑜2) Θ(𝑜3) Arbitrary 𝑈 Average ball size: Θ(𝑜2.5) Explicit formulas.

Radius 𝑢 (fixed)

23

𝑈 𝑈 𝑈 Θ(𝑜2𝑢) Θ(𝑜3𝑢) Arbitrary 𝑈 Average ball size: Θ(𝑜2.5𝑢) Recursive formulas.

The Results of this Work

• Correcting 𝒐 − 𝟑 erasures: 𝐵 𝑜, 𝑜 − 1 = 𝑜/2 .
• Correcting 𝒐 − 𝟒 erasures: 𝐵 𝑜, 𝑜 − 2 = 𝑜.
• Correcting 𝒐 − 𝟓 erasures: 𝐵 𝑜, 𝑜 − 3 = 𝑃 𝑜2 .
• For fixed 𝑒 and 𝑜 ≥ 2𝑒,

Ω 𝑜𝑜−2𝑒 ≤ 𝐵 𝑜, 𝑒 ≤ 𝑃 𝑜𝑜−1−𝑒 .

• Studying tree balls of trees.

24

Conclusions and Future Work

• Improve the lower and upper bounds on 𝐵(𝑜, 𝑒).
• Study codes over trees under different metrics such as the tree edit

distance.

• Study the problem of reconstructing trees based upon several forests

in the forest ball of trees.

25

Thank You!